Theorem crnggrpd 20219

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20218	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20214	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2119  Grpcgrp 18900  CRingccrg 20206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-ring 20207  df-cring 20208

This theorem is referenced by:  fermltlchr  21504  selvvvval  22118  selvadd  22119  psdmul  22154  psd1  22155  psdpw  22158  ply1chr  22292  ply1fermltlchr  22298  elrgspnsubrunlem1  33328  elrgspnsubrunlem2  33329  erlbr2d  33345  rlocaddval  33349  rloccring  33351  rloc0g  33352  rlocf1  33354  fracerl  33390  gsumind  33428  ressply1evls1  33648  evl1deg1  33659  evl1deg2  33660  evl1deg3  33661  ply1dg1rt  33663  vr1nz  33676  mplasclco  33700  psrmonprod  33736  mplmonprod  33738  esplyfvn  33761  irngss  33871  extdgfialglem1  33876  irredminply  33900  algextdeglem4  33904  algextdeglem5  33905  aks6d1c1p3  42595  aks6d1c2lem4  42612  aks6d1c6lem2  42656  aks5lem2  42672  evlsbagval  43036  evlselv  43039  prjcrv0  43083